Theorem cnconst 23657

Description: A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
cnconst	⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵 ∈ 𝑌 ∧ 𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾))

Proof of Theorem cnconst

Step	Hyp	Ref	Expression
1		fconst2g 7240	. . . 4 ⊢ (𝐵 ∈ 𝑌 → (𝐹:𝑋⟶{𝐵} ↔ 𝐹 = (𝑋 × {𝐵})))
2	1	adantl 487	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵 ∈ 𝑌) → (𝐹:𝑋⟶{𝐵} ↔ 𝐹 = (𝑋 × {𝐵})))
3		cnconst2 23656	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
4	3	3expa 1136	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
5		eleq1 2852	. . . 4 ⊢ (𝐹 = (𝑋 × {𝐵}) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾)))
6	4, 5	syl5ibrcom 250	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵 ∈ 𝑌) → (𝐹 = (𝑋 × {𝐵}) → 𝐹 ∈ (𝐽 Cn 𝐾)))
7	2, 6	sylbid 243	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐵 ∈ 𝑌) → (𝐹:𝑋⟶{𝐵} → 𝐹 ∈ (𝐽 Cn 𝐾)))
8	7	impr 460	1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵 ∈ 𝑌 ∧ 𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {csn 4588   × cxp 5656  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  TopOnctopon 23283   Cn ccn 23597

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887  df-topgen 17664  df-top 23267  df-topon 23284  df-cn 23600  df-cnp 23601

This theorem is used by:  xrge0mulc1cn  34684  cxpcncf2  47182